Check the injectivity and surjectivity of the following functions:

• (i) f: N → N given by f(x) = x²
• (ii) f: Z → Z given by f(x) = x²
• (iii) f: R → R given by f(x) = x²
• (iv) f: N → N given by f(x) = x³
• (v) f: Z → Z given by f(x) = x³

To determine the injectivity and surjectivity of the given functions, we will analyze each one individually.

1. Function: f: N → N, f(x) = x²

Injectivity: This function is not injective because different natural numbers can yield the same output. For example, f(2) = 4 and f(-2) = 4 (though -2 is not in N, it illustrates the point).

Surjectivity: This function is not surjective either, as not every natural number is a perfect square. For instance, there is no natural number x such that f(x) = 3.

2. Function: f: Z → Z, f(x) = x²

Injectivity: This function is not injective. For example, f(2) = 4 and f(-2) = 4, showing that two different integers map to the same output.

Surjectivity: It is also not surjective because negative integers cannot be expressed as squares of any integer. For instance, there is no integer x such that f(x) = -1.

3. Function: f: R → R, f(x) = x²

Injectivity: This function is not injective. Similar to the previous cases, f(2) = 4 and f(-2) = 4 demonstrate that different real numbers can yield the same result.

Surjectivity: It is not surjective as well, since negative real numbers cannot be achieved. There is no real number x such that f(x) = -1.

4. Function: f: N → N, f(x) = x³

Injectivity: This function is injective. If f(a) = f(b), then a³ = b³ implies a = b, meaning different inputs yield different outputs.

Surjectivity: It is not surjective because not every natural number is a perfect cube. For example, there is no natural number x such that f(x) = 2.

5. Function: f: Z → Z, f(x) = x³

Injectivity: This function is injective. If f(a) = f(b), then a³ = b³ implies a = b, ensuring unique outputs for unique inputs.

Surjectivity: It is surjective because every integer can be expressed as the cube of some integer. For any integer y, there exists an integer x such that f(x) = y.